[1]} introduced the idea that an unconstrained model can be seen as a parameter expansion of a constrained model and applyed it to latent factor models and, more in general, hierarchical models with latent variables.
Following this idea, for the purpose of the sampling algorithm, we start from the unconstrained DPM for the distribution of the innovations,
which is a parameter expanded (in the sense of [2]}, [3]}
and [4]}) version of the DPMLN2-vMEM in (REF ) and will
be called PX-DPMLN2-vMEM. It is important to enlighten that a prior
on the parameters of the PX model induces a prior on the parameters
of the original model and that the use of proper priors results in
proper posteriors for this model (even if the likelihood is improper).
